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1 FEE Download Stud Package from website: wwwtekoclassescom & wwwmathsbsuhagcom Get Solution of These Packages & Learn b Video Tutorials on wwwmathsbsuhagcom Matrices An rectangular arrangement of numbers (real or comple) (or of real valued or comple valued epressions) is called a matri If a matri has m rows and n columns then the order of matri is said to be m b n (denoted as m n) The general m n matri is a a a aj an a a a aj an A = ai ai ai aij ain am am am amj amn where a ij denote the element of i th row & j th column The above matri is usuall denoted as [a ij Note : (i) The elements a, a, a, are called as diagonal elements Their sum is called as trace of A denoted as T r (A) (ii) Capital letters of English alphabets are used to denote a matri Basic Definitions (i) ow matri : A matri having onl one row is called as row matri (or row vector) General form of row matri is A = [a, a, a,, a n ] (ii) Column matri : A matri having onl one column is called as column matri (or column vector) a Column matri is in the form A = a am (iii) Square matri : A matri in which number of rows & columns are equal is called a square matri General form of a square matri is a a an A = a a an an an ann which we denote as A = [a ij (iv) Zero matri : A = [a ij is called a ero matri, if a ij = i & j (v) Upper triangular matri : A = [a ij is said to be upper triangular, if a ij = for i > j (ie, all the elements below the diagonal elements are ero) (vi) Lower triangular matri : A = [a ij is said to be a lower triangular matri, if a ij = for i < j (ie, all the elements above the diagonal elements are ero) (vii) Diagonal matri : A square matri [a ij is said to be a diagonal matri if a ij = for i j (ie, all the elements of the square matri other than diagonal elements are ero) (viii) Note : Diagonal matri of order n is denoted as Diag (a, a, a nn ) Scalar matri :Scalar matri is a diagonal matri in which all the diagonal elements are same A = [a ij is a scalar matri, if (i) a ij = for i j and (ii) a ij = k for i = j (i) Unit matri (Identit matri) : Unit matri is a diagonal matri in which all the diagonal elements are unit Unit matri of order 'n' is denoted b n (or ) ie A = [a ij is a unit matri when a ij = for i j & a ii = eg =, = () Comparable matrices : Two matrices A & B are said to be comparable, if the have the same order (ie, number of rows of A & B are same and also the number of columns) (i) Equalit of matrices : Two matrices A and B are said to be equal if the are comparable and all the corresponding elements are equal & B = [b ij ] p q A = B iff (i) m = p, n = q (ii) a ij = b ij i & j (ii) Multiplication of matri b scalar : Let be a scalar (real or comple number) & A = [a ij be a matri Thus the product A is defined as A = [b ij where b ij = a ij i & j Note : If A is a scalar matri, then A =, where is the diagonal element (iii) Addition of matrices : Let A and B be two matrices of same order (ie comparable matrices) Successful People eplace the words like; "wish", "tr" & "should" with "I Will" Ineffective People don't Teko Classes, Maths : Suhag Karia (S K Sir), Bhopal, Phone : , page of 5

2 FEE Download Stud Package from website: wwwtekoclassescom & wwwmathsbsuhagcom Get Solution of These Packages & Learn b Video Tutorials on wwwmathsbsuhagcom Then A + B is defined to be A + B = [a ij + [b ij (iv) = [c ij where c ij = a ij + b ij i & j Substraction of matrices : Let A & B be two matrices of same order Then A B is defined as A + B where B is ( ) B (v) Properties of addition & scalar multiplication : Consider all matrices of order m n, whose elements are from a set F (F denote Q, or C) Let M m n (F) denote the set of all such matrices Then (a) A M m n (F) & B M m n (F) A + B M m n (F) (vi) (b) A + B = B + A (c) (A + B) + C = A + (B + C) (d) O = [o is the additive identit (e) For ever A M m n (F), A is the additive inverse (f) (A + B) = A + B (g) A = A (h) ( + ) A = A + A Multiplication of matrices : Let A and B be two matrices such that the number of columns of A is same as number of rows of B ie, A = [a ij ] m p & B = [b ij ] p n Then AB = [c ij where c ij = a column vector of B p ik b kj k, which is the dot product of i th row vector of A and j th Note - : The product AB is defined iff number of columns of A equals number of rows of B A is called as premultiplier & B is called as post multiplier AB is defined / BA is defined Note - : In general AB BA, even when both the products are defined Note - : A (BC) = (AB) C, whenever it is defined (vii) Properties of matri multiplication : Consider all square matrices of order 'n' Let M n (F) denote the set of all square matrices of order n (where F is Q, or C) Then (a) A, B M n (F) AB M n (F) (b) In general AB BA (c) (AB) C = A(BC) (d) n, the identit matri of order n, is the multiplicative identit A n = A = n A A M n (F) (e) For ever non singular matri A (ie, A ) of M n (F) there eist a unique (particular) matri B M n (F) so that AB = n = BA In this case we sa that A & B are multiplicative inverse of one another In notations, we write B = A or A = B (f) If is a scalar (A) B = (AB) = A(B) (g) A(B + C) = AB + AC A, B, C M n (F) (h) (A + B) C = AC + BC A, B, C M n (F) Note : (i) Then A n = A & m A = A, where n & m are identit matrices of order n & m respectivel (ii) For a square matri A, A denotes AA, A denotes AAA etc Solved Eample # Solution Let A = sin / cos / / cos & B = cos tan cos sin cos Find so that A = B B definition A & B are equal if the have the same order and all the corresponding elements are equal Thus we have sin =, cos = & tan = = (n + ) Solved Eample # f() is a quadratic epression such that a a f() a b b f() = b for three unequal numbers a, b, c Find f() c c f( ) c Solution The given matri equation implies a f() af() f( ) a b f() bf() f( ) = b c f() cf() f( ) c f() + f() + f( ) = + for three unequal numbers a, b, c (i) Teko Classes, Maths : Suhag Karia (S K Sir), Bhopal, Phone : , page of 5 Successful People eplace the words like; "wish", "tr" & "should" with "I Will" Ineffective People don't

3 FEE Download Stud Package from website: wwwtekoclassescom & wwwmathsbsuhagcom Get Solution of These Packages & Learn b Video Tutorials on wwwmathsbsuhagcom (i) is an identit f() =, f() = & f( ) = = a + b & = a + b f() = (a + b) b = & a = f() = + Self Practice Problems : cos sin If A() =, varif that A() A() = A( + ) sin cos Hence show that in this case A() A() = A() A() Let A =, B = and C = [ ] 5 Then which of the products ABC, ACB, BAC, BCA, CAB, CBA are defined Calculate the product whichever is defined Ans onl CAB is defined CAB = [5 ] Transpose of a Matri Let A =[a ij Then the transpose of A is denoted b A( or A T ) and is defined as A = [b ij m where b ij = a ji i & j ie A is obtained b rewriting all the rows of A as columns (or b rewriting all the columns of A as rows) (i) For an matri A = [a ij, (A) = A (ii) Let be a scalar & A be a matri Then (A) = A (iii) (A + B) = A + B & (A B) = A B for two comparable matrices A and B (iv) (A ± A ± ± A n ) = A ± A ± ± A n, where A i are comparable (v) ] m p & B = [b ij ] p n, then (AB) = BA (vi) (A A A n )= A n A n A A, provided the product is defined (vii) Smmetric & skew smmetric matri : A square matri A is said to be smmetric if A = A ie A is smmetric iff a ij = a ji i & j A square matri A is said to be skew smmetric if A = A ie A is skew smmetric iff a ij = a ji i & j a h g eg A = h b f is a smmetric matri g f c o B = o is a skew smmetric matri Note- In a skew smmetric matri all the diagonal elements are ero ( a ii = a ii a ii = ) Note- For an square matri A, A + A is smmetric & A A is skew smmetric Note- Ever square matri can be uniqual epressed as sum of two square matrices of which one is smmetric and other is skew smmetric A = B + C, where B = (A + A) & C = (A A) Solved Eample # Show that BAB is smmetric or skew smmetric according as A is smmetric or skew smmetric (where B is an square matri whose order is same as that of A) Solution Case - A is smmetric A = A (BAB) = (B)AB = BAB BAB is smmetric Case - A is skew smmetric A = A (BAB) = (B)AB = B ( A) B = (BAB) BAB is skew smmetric Self Practice Problems : For an square matri A, show that AA & AA are smmetric matrices If A & B are smmetric matrices of same order, than show that AB + BA is smmetric and AB BA is skew smmetric Submatri, Minors, Cofactors & Determinant of a Matri (i) Submatri : Let A be a given matri The matri obtained b deleting some rows or columns of A is called as submatri of A a b c d eg A = w p q r s a c a b c a b d Then,, p r p q s are all submatrices of A p q r Successful People eplace the words like; "wish", "tr" & "should" with "I Will" Ineffective People don't Teko Classes, Maths : Suhag Karia (S K Sir), Bhopal, Phone : , page of 5

4 FEE Download Stud Package from website: wwwtekoclassescom & wwwmathsbsuhagcom Get Solution of These Packages & Learn b Video Tutorials on wwwmathsbsuhagcom (ii) Determinant of a square matri : Let A = [a] be a matri Determinant A is defined as A = a eg A = [ ] A = a b Let A =, then A is defined as ad bc c d 5 eg A =, A = (iii) Minors & Cofactors : be a square matri Then minor of element a ij, denoted b M ij is defined as the determinant of the submatri obtained b deleting i th row & j th column of A Cofactor of element a ij, denoted b C ij (or A ij ) is defined as C ij = ( ) i + j M ij a b eg A = c d M = d = C M = c, C = c M = b, C = b M = a = C a b c eg A = p q r q r M = = q r = C M = a b = a b, C = (a b) = b a etc (iv) Determinant of an order : be a square matri (n > ) Determinant of A is defined as the sum of products of elements of an one row (or an one column) with corresponding cofactors a a a eg A = a a a a a a A = a C + a C + a C (using first row) (v) (vi) (vii) a a a a a a = a a a a + a a a a a A = a C + a C + a C (using second column) a a a a a a = a + a a a a a a a a Some properties of determinant (a) A = A for an square matri A (b) If two rows are identical (or two columns are identical) then A = (c) (d) (e) Let be a scalar Than A is obtained b multipling an one row (or an one column) of A b Note : A = n A, when A = [a ij The sum of the products of elements of an row with corresponding cofactors of an other row is ero (Similarl the sum of the products of elements of an column with corresponding cofactors of an other column is ero) If A and B are two square matrices of same order, then AB = A B Note : As A = A, we have A B = AB (row - row method) A B = AB (column - column method) A B = AB (column - row method) Singular & non singular matri : A square matri A is said to be singular or non singular according as A is ero or non ero respectivel Cofactor matri & adjoint matri : be a square matri The matri obtained b replacing each element of A b corresponding cofactor is called as cofactor matri of A, denoted as cofactor A The transpose of cofactor matri of A is called as adjoint of A, denoted as adj A Teko Classes, Maths : Suhag Karia (S K Sir), Bhopal, Phone : , page 5 of 5 Successful People eplace the words like; "wish", "tr" & "should" with "I Will" Ineffective People don't

5 Get Solution of These Packages & Learn b Video Tutorials on wwwmathsbsuhagcom ie if A = [a ij then cofactor A = [c ij when c ij is the cofactor of a ij i & j Adj A = [d ij where d ij = c ji i & j FEE Download Stud Package from website: wwwtekoclassescom & wwwmathsbsuhagcom (viii) Properties of cofactor A and adj A: (a) A adj A = A n = (adj A) A where A = [a ij (b) adj A = A n, where n is order of A In particular, for matri, adj A = A (c) If A is a smmetric matri, then adj A are also smmetric matrices (d) If A is singular, then adj A is also singular (i) emarks : Inverse of a matri (reciprocal matri) : Let A be a non singular matri Then the matri adj A is the multiplicative inverse of A (we call it inverse of A) and is denoted b A A We have A (adj A) = A n = (adj A) A A adj A = A n = adj A A, for A is non singular A A = adj A A The necessar and sufficient condition for eistence of inverse of A is that A is non singular A is alwas non singular If A = dia (a, a,, a nn ) where a ii i, then A = diag (a, a,, a nn ) (A ) = (A) for an non singular matri A Also adj (A) = (adj A) 5 (A ) = A if A is non singular Let k be a non ero scalar & A be a non singular matri Then (ka) = k A 7 A = A for A 8 Let A be a nonsingular matri Then AB = AC B = C & BA = CA B= C 9 A is non-singular and smmetric A is smmetric In general AB = does not impl A = or B = But if A is non singular and AB =, then B = Similarl B is non singular and AB = A = Therefore, AB = either both are singular or one of them is Solved Eample # For a skew smmetric matri A, show that adj A is a smmetric matri Solution A = a b a c b c adj A = (cof A) = c bc ca bc b ab c bc ca cof A = bc b ab ca ab a ca ab which is smmetric a Solved Eample # 5 For two nonsingular matrices A & B, show that adj (AB) = (adj B) (adj A) Solution We have (AB) (adj (AB)) = AB n= A B n A (AB)(adj (AB)) = A B A B adj (AB) = B adj A ( A = B B adj (AB) = B B adj A adj (AB) = (adjb) (adj A) A adj A) Teko Classes, Maths : Suhag Karia (S K Sir), Bhopal, Phone : , page of 5 Successful People eplace the words like; "wish", "tr" & "should" with "I Will" Ineffective People don't

6 FEE Download Stud Package from website: wwwtekoclassescom & wwwmathsbsuhagcom Get Solution of These Packages & Learn b Video Tutorials on wwwmathsbsuhagcom Self Practice Problems : If A is nonsingular, show that adj (adj A) = A n A Prove that adj (A ) = (adj A) (n) For an square matri A, show that adj (adj A) = A If A and B are nonsingular matrices, show that (AB) = B A Sstem of Linear Equations & Matrices Consider the sstem a + a + + a n n = b a + a + + a n n = b a m + a m + + a mn n = b n b a a an Let A = a a a n, X = b & B = am am amn n bn Then the above sstem can be epressed in the matri form as AX = B The sstem is said to be consistent if it has atleast one solution (i) Sstem of linear equations and matri inverse: If the above sstem consist of n equations in n unknowns, then we have AX = B where A is a square matri If A is nonsingular, solution is given b X = A B If A is singular, (adj A) B = and all the columns of A are not proportional, then the sstem has infinite man solution If A is singular and (adj A) B, then the sstem has no solution (we sa it is inconsistent) (ii) Homogeneous sstem and matri inverse: If the above sstem is homogeneous, n equations in n unknowns, then in the matri form it is AX = O ( in this case b = b = b n = ), where A is a square matri If A is nonsingular, the sstem has onl the trivial solution (ero solution) X = If A is singular, then the sstem has infinitel man solutions (including the trivial solution) and hence it has non trivial solutions (iii) ank of a matri : ] m n A natural number is said to be the rank of A if A has a nonsingular submatri of order and it has no nonsingular submatri of order more than ank of ero matri is regarded to be ero 5 eg A = 5 we have as a non singular submatri The square matrices of order are 5 5 5,,, and all these are singular Hence rank of A is (iv) Elementar row transformation of matri : The following operations on a matri are called as elementar row transformations (a) Interchanging two rows (b) Multiplications of all the elements of row b a nonero scalar (c) Addition of constant multiple of a row to another row Note : Similar to above we have elementar column transformations also emark : Elementar transformation on a matri does not affect its rank Two matrices A & B are said to be equivalent if one is obtained from other using elementar transformations We write A B (v) Echelon form of a matri : A matric is said to be in Echelon form if it satisf the followings: (a) The first non-ero element in each row is & all the other elements in the corresponding column (ie the column where appears) are eroes (b) The number of eroes before the first non ero element in an non ero row is less than the number of such eroes in succeeding non ero rows esult : ank of a matri in Echelon form is the number of non ero rows (ie number of rows with atleast one non ero element) emark : To find the rank of a given matri we ma reduce it to Echelon form using elementar row transformations and then count the number of non ero rows Successful People eplace the words like; "wish", "tr" & "should" with "I Will" Ineffective People don't Teko Classes, Maths : Suhag Karia (S K Sir), Bhopal, Phone : , page 7 of 5

7 Successful People eplace the words like; "wish", "tr" & "should" with "I Will" Ineffective People don't Teko Classes, Maths : Suhag Karia (S K Sir), Bhopal, Phone : , page 8 of 5 FEE Download Stud Package from website: wwwtekoclassescom & wwwmathsbsuhagcom Get Solution of These Packages & Learn b Video Tutorials on wwwmathsbsuhagcom (vi) Sstem of linear equations & rank of matri: Let the sstem be AX = B where A is an m n matri, X is the n-column vector & B is the m-column vector Let [AB] denote the augmented matri (ie matri obtained b accepting elements of B as n + th column & first n columns are that of A) (A) denote rank of A and ([AB]) denote rank of the augmented matri Clearl (A) ([AB]) (a) If (A) < ([AB]) then the sstem has no solution (ie sstem is inconsistent) (b) If (A) = ([AB]) = number of unknowns, then the sstem has unique solution (and hence is consistent) (c) If (A) = ([AB]) < number of unknowns, then the sstems has infinitel man solutions (and so is consistent) (vii) Homogeneous sstem & rank of matri : Let the homogenous sstem be AX =, m equations in 'n' unknowns In this case B = and so (A) = ([AB]) Hence if (A) = n, then the sstem has onl the trivial solution If (A) < n, then the sstem has infinitel man solutions Solved Eample # Solve the sstem using matri inverse Solution Let A =, X = & B = Then the sstem is AX = B A = Hence A is non singular Cofactor A = adj A = A = A adj A = = / / / / / / / X = A B = / / / / / / / ie = =, =, = Solved Eample # 7 Test the consistanc of the sstem 8 Also find the solution, if an Solution A = X =, B = 8 [AB] = 8

8 FEE Download Stud Package from website: wwwtekoclassescom & wwwmathsbsuhagcom Get Solution of These Packages & Learn b Video Tutorials on wwwmathsbsuhagcom / / / / / This is in Echelon form (AB) = = (A) < number of unknowns Hence there are infinitel man solutions n = Hence we can take one of the variables an value and the rest in terms of it Let = r, where r is an number Then = r + = r r r = & = r r Solutions are (,, ) =,,r Self Practice Problems: A = Find the inverse of A using A and adj A Also find A b solving a sstem of equations Find real values of and µ so that the following sstems has (i) unique solution + + = (ii) infinite solution (iii) No solution + + = + + = µ Ans (i), µ (ii) =, µ = (iii) =, µ Find so that the following homogeneous sstem have a non ero solution + + = + + = + + = Ans = 5 More on Matrices (i) Characteristic polnomial & Characteristic equation : Let A be a square matri Then the polnomial A is called as characteristic polnomial of A & the equation A = is called as characteristic equation of A emark : Ever square matri A satisf its characteristic equation (Cale - Hamilton Theorem) ie a n + a, n + + a n + a n = is the characteristic equation of A, then a A n + a A n + + a n A + a n = (ii) More Definitions on Matrices : (a) Nilpotent matri: A square matri A is said to be nilpotent ( of order ) if, A = O A square matri is said to be nilpotent of order p, if p is the least positive integer such that A p = O (b) Idempotent matri: A square matri A is said to be idempotent if, A = A Teko Classes, Maths : Suhag Karia (S K Sir), Bhopal, Phone : , page 9 of 5 Successful People eplace the words like; "wish", "tr" & "should" with "I Will" Ineffective People don't

9 FEE Download Stud Package from website: wwwtekoclassescom & wwwmathsbsuhagcom Get Solution of These Packages & Learn b Video Tutorials on wwwmathsbsuhagcom (c) (d) (e) eg is an idempotent matri Involutor matri: A square matri A is said to be involutor if A =, being the identit matri eg A = is an involutor matri Orthogonal matri: A square matri A is said to be an orthogonal matri if, A A = = AA Unitar matri: A square matri A is said to be unitar if A( A ) =, where A is the comple conjugate of A Solved Eample # 8 If A =, show that 5A = A + A 5 Solution We have the characteristic equation of A A = ie = ie = Using cale - Hamilton theorem A + A 5A 5 = 5 = A + A 5A Multipling b A, we get 5A = A + A 5 Solved Eample # 9 Show that a square matri A is involutor, iff ( A) ( + A) = Solution Let A be involutor Then A = ( A) ( + A) = + A A A = + A A A = A = Conversl, let ( A) ( + A) = + A A A = + A A A = A = A is involutor Self Practice Problems If A is idempotent, show that B = A is idempotent and that AB = BA = If A is a nilpotent matri of inde, show that A ( + A) n = A for all n N A is a skew smmetric matri, such that A + = Show that A is orthogonal and is of even order c b Let A = c a If A + A =, find b a Ans a + b + c Teko Classes, Maths : Suhag Karia (S K Sir), Bhopal, Phone : , page of 5 Successful People eplace the words like; "wish", "tr" & "should" with "I Will" Ineffective People don't